Abstract. A transient chaos in a closed FRW cosmological model with a scalar field is studied. We describe two different chaotic regimes and show that the type of chaos in this model depends on the scalar field potential. We have found also that for sufficiently steep potentials or for potentials with large cosmological constant the chaotic behavior disappears.
Key words: cosmology; scalar field; chaotic dynamics
2000 Mathematics Subject Classification: 37B25; 83C75
1
Introduction
The phenomenon of a transient chaos (when a dynamical system behaves chaotically during a transient period before reaching some other regime) becomes in last two decade a matter of intense investigations. More than 20 years ago it was remarked that a periodic attractor may follow a temporal chaotic behavior (see [1] and references therein). After, this kind of chaos has been found in a broader class of dynamical systems which have no attractors at all. In the paper of Gaspard and Rice [2] the dynamics generated by scattering of a small disk particle on three hard discs in a plane was described. For an arbitrary initial data apart from a zero measure set this particle leaves the system after some time interval (which can be arbitrarily large), during which the small disc has a chain of scattering on the three discs.
Several years earlier this kind of behavior was discovered in a cosmological model describing the evolution of a closed Friedman–Robertson–Walker (FRW) Universe filled with a massive scalar field [3]. The analog of scattering on a disc in this model is “bounce” – a transition from a cosmological collapse to a cosmological expansion of the Universe. The final regime of the dynamics for almost all initial condition is falling into a cosmological singularity. The description of the set of periodic trajectories in this model in the language of symbolic dynamics and calculation of topological entropy have been done in [4]. Later, such analysis was done for cosmological models with other types of a scalar field [5, 6, 7]. It appears that depending on the particular form of the scalar field potential, the dynamics may be either chaotic or regular. Several types of transition from chaos to a non-chaotic dynamics for particular one-parameter families of potential were described in [7]. In the present paper we summarize our knowledge on possible regimes in closed FRW cosmology with a scalar field.
2
Equations of motion and basic properties
We study the following ODE system (the derivation see in [7]):
m2
P 16π
¨ a+ a˙
2
2a + 1 2a
+aϕ˙
2
8 − aV(ϕ)
¨ ϕ+3 ˙ϕa˙
a +V
′(ϕ) = 0. (2)
with two variables – a scale factoraand a scalar fieldϕ. HeremP is a constant fixed parameter – the Planck mass, the scalar field potentialV(φ) is a smooth nonnegative function withV(0) = 0. The ratio H ≡a/a˙ is called the Hubble parameter.
This system has one first integral of motion
−8π3 m2Pa a˙2+ 1 + a
3
2 ϕ˙
2+ 2V(ϕ)
=C. (3)
This integral play the role of energy and is equal to zero for cosmological solutions of the sys-tem (1), (2). This property arises from the known fact that in General Relativity the Hamiltonian vanishes identically. That is why in this section we restrict ourself by C = 0 three-dimensional hypersurface in the whole 4-dimensional phase space of (1), (2) leaving the general situation to the Section 4.
The equation (2) for the scalar field looks like the equation for a harmonic oscillator with a time dependent “friction” 3Hwhich is positive in an expanding Universe. One of the most im-portant in modern cosmology, the slow-roll regime can occurs when this “friction” is much larger than the frequency of the oscillator. More precisely, slow-roll approximation is characterized by the system
H = s
8π 3m2
P
V(ϕ), ϕ˙ = V ′(ϕ) 3H ,
resulting by neglecting second derivative terms, kinetic energy of the scalar field and spatial curvature. This regime in rather natural for physically admissible initial conditions [9, 10, 11] and leads to fast growth of the scale factor a while the scalar field ϕ slow rolls toward zero. When the scalar field ϕfalls below some value ϕ∼mP this regime disappears [8].
In the opposite case, when the “friction” is small, the dynamics of ϕ is damping oscilla-tions [12]. This regime is typical for late time evolution of the Universe. However, in contrast to zero- or negative spatial curvature cases, for a Universe with a positive spatial curvature this regime is not its finite fate, and is ultimately followed by a recollapse of the Universe.
Figure 1. The boundary (5) (solid curves) for the potentialV =m2 ϕ2
/2. Points of maximal expansion are located between two dashed curves while bounces are possible in two narrow zones between solid and dashed curves.
3
The nature of chaotic regime
A very important property of the constraint equation (3) is that ˙a2and ˙ϕ2 enter in the left-hand side with opposite signs. Rewriting (3) in the form
−3m
2
P 8π
˙ a2
a2 +
˙ ϕ2
2 = 3m2P
8π 1
a2 −V(ϕ) (4)
it is easy to see that
• There are no forbidden regions in the configuration space (a, ϕ).
• The configuration space is divided into two regions – the region where the right-hand side of (4) is positive (and possible extrema of the scale factor are located), and the region where it is negative (where possible extrema of the scalar field are located).
The boundary between these two regions is the curve
a2= 3 8π
m2P
V(ϕ). (5)
Zero-velocity points ( ˙a= ˙ϕ= 0) of a trajectory, if they exist, should lie on this curve.
Consider points of maximal expansion and those of minimal contraction, i.e. the points, where ˙
a= 0 in more detail. They can exist only in the region where
a2≤ 3 8π
m2
P
V(ϕ), (6)
Using (1), we can show that possible points of maximal expansion ( ˙a= 0, ¨a <0) are localized inside the region
a2≤ 1 4π
m2
P
V(ϕ) (7)
while possible points of minimal contraction ( ˙a= 0, ¨a >0) lie outside this region (7) being at the same time inside the region (6) (see Fig. 1).
such that a trajectory, starting from them after some finite time reaches a point a= 0,ϕ=∞. In this point the equations of motion become singular and the trajectory can not be continued further in time. In General Relativity, this situation is called as a cosmological singularity. For bigger initial values of the initial scale factor one can see a narrow region where a trajectory experiences a bounce i.e. goes through the point of minimal contraction, after that the scale factor aalong the trajectory begins to grow. Increasing initial a further, we obtain the region where a trajectory has a “ϕ-turn” i.e. has the extremum of the scalar field ϕ and then falls into singularity. Then one has a region corresponding to trajectories having bounce after one oscillation in ϕand so on.
To avoid a misunderstanding, let us indicate once more what initial condition space we use. When we start from maximal expansion point, we fix one time derivative ( ˙a= 0). So, to fix the initial condition completely, we need only to specify initial values of a and ϕ. The initial ˙ϕ is determined from the constraint equation (3), and our initial condition space becomes (a, ϕ). We will study the structure of this space, investigating the location of points of maximal expansion, starting from which a trajectory has a bounce, but not the location of bounce itself. The ϕ= 0 cross-section of regions, leading to bounce, are called as “bounce intervals”.
Remembering that a trajectory describing an expanding universe must have a point of maxi-mal expansion, we can apply this analysis further to bouncing trajectories after a bounce. Their second point of maximal expansion may lie either inside bounce regions, or between them. This fact generates a substructure of the region under consideration. The structure of subregions leading to two bounces repeats in general the structure of regions having at least one bounce, and so on and so forth. Continuing this process ad infinitum, we get the fractal zero-measure set of infinitely bouncing trajectories escaping the singularity.
If we distinguish two possible singular outcomes, ϕ → +∞ and ϕ → −∞, the method of basins boundaries can also be applied. Suppose that initial velocity in the point of maximal expansion is directed “up” (initial ˙ϕ > 0). Then trajectories with even number of ϕ-turns will approach ϕ→+∞ singular point, while trajectories with odd number of ϕ-turns fall into ϕ→ −∞ singularity. A boundary between these two basins is fractal, indicating the presence of chaos (see numerical examples in [4]).
In the first paper on chaos in FRW cosmology [3] D. Page have used another description. Instead of starting from a maximum expansion point, he began the analysis from zero-velocity point at the curve (5). Depending on the initial point, two different situations can be distin-guished. Trajectories, going from this curve into the region (6) (the space between two solid hyperbolae in Fig. 1) have a point of maximal expansion soon after start, and then go towards a singularity. So, to prevent an almost immediate collapse, a trajectory must be directed into the region, where extrema of the scalar field are located (outside a solid hyperbola in Fig. 1). These two situations are separated by a particular trajectory, tangent to the curve (5). It means that this trajectory has
¨ ϕ ¨ a =
dϕ da
at the initial point. Hereϕ(a) in the right-hand side is the equation of the curve (5). This point was first introduced by Page in [3] for massive scalar field potential V(φ) = m2φ2/2. In this
case
ϕpage =
r 3
4πmP, apage = 1/m, except for the trivial solutionϕ= 0,a=∞.
If at the initial point on the boundary (5)
¨ φ ¨ a <
dϕ
Figure 2. Example of trajectories with the initial conditions close to the boundary separating trajectories falling intoϕ= +∞(trajectory 1) andϕ=−∞(trajectories 2–5) singularities for the caseϕ0<
√3
4√πmP. This boundary is sharp, no fractal structure is present. Trajectories 2–5 have a zigzag-like form, no periodical trajectories are present. The long-dashed line is the boundary (5), the short-dashed line separates zones of bounces from zones of maximal expansion points.
so that a trajectory is directed outside of the region (6), the Universe experiences a long enough expansion phase. For a massive scalar field (8) is satisfied ifϕ >(3/4π)mP. It should be noted, thata priorithe significance of such trajectories for the chaotic structure is not evident, because we study now only those with a zero-velocity point. However, numerical data show that a large class of chaotic trajectories has zero-velocity points; in particular, all primary periodical orbits (i.e. having one bounce per period) contain such points (see examples in [4]).
With use of the equation of motion (1), (2) the criterion (8) gives for an arbitrary scalar field potential
V(ϕ)> r
3m2P 16π V
′(ϕ). (9)
The condition (9) may be treated as a restriction of local steepness of the function V(ϕ). In can be easily seen that for power-law potentials there exists a value ϕpage such that for
all ϕ > ϕpage the inequality (9) is satisfied. For steeper potentials the situation changes. For
example, the potentialV(ϕ) =M04(cosh(ϕ/ϕ0)−1) has a Page point only ifϕ0 >
√
3
4√πmP. In the opposite case the condition (9) is never satisfied and all trajectories starting from the curve (5) go into the region (6), experience the point of maximal expansion and fall into a singularity. Periodical trajectories with a zero-velocity point can not exist. It has been confirmed numerically that the chaos is absent in this case (periodical trajectories disappear, and the boundary between basins of attractions of ϕ → +∞ and ϕ → −∞ singularities becomes smooth) [7]. Although bounces with ˙a = 0, ¨a > 0 are still possible, trajectories with bounces have a zigzag form, do not exit the region (6), and restore their direction to a singularity soon after the bounce (see Fig. 2).
For steeper potentials like
V(ϕ) =M04 exp ϕ2/ϕ20
+ exp −ϕ2/ϕ20 −2
the condition (9) is definitely violated for large ϕbut, depending on ϕ0 it can be satisfied for
0 5 10 15 20 25 30 a
-100 -75 -50 -25 0
C
Figure 3. Theϕ= 0 cross-section of the bounce intervals for the potentialV =m2 ϕ2
/2 and negativeC. Consecutive merging of 5 first intervals can be seen in this range ofC.
simulations show that the chaotic behavior exists ifϕ0 >0.96mP. So, we have a rather accurate and easily calculable condition for existence of the chaotic dynamics in the system (1), (2) (another example of a very steep potential see in [7]).
4
Non-zero
C
Now we return to general properties of (1), (2) with an arbitrary value of the energy integralC in (3). One particular case of this problem have already been investigated. Namely, it can be shown that our dynamical system withpositiveCif formally equivalent to the system describing a scalar field in the presence of a pressure-less matter. A detailed study of this case for massive scalar field potential V(φ) = m2ϕ2/2 have been done in [6]. The structure of the periodical
trajectories becomes more complicated in comparison withC = 0 case. First of all, the number of bounce interval becomes finite and diminishes with increasingC. Besides, we have a nontrivial set of selection rules, which should be satisfied for any allowed sequence of intervals along an arbitrary trajectory (in general, the bigger is the ordinal number of the first interval, the smaller should be a maximum possible number of the second one). One particular example of such rules (simplified a little in comparison with those seen in computer simulations) was described in [6] with calculation of corresponding topological entropy. If C increases, the topological entropy decreases (the width of initial space regions leading to bounce decreases also), and for Cm >0.023m2
P the chaos disappears.
For positiveC this situation is general – chaos disappears for energy integral exceeding some critical value which depends on particular form of the potential. A negativeC alter the nature of chaos in a different way. Though there are no direct physical applications of a system (1), (2) with a negative C, mathematical properties of this case are very interesting. Besides, there are dynamical systems of the form similar to (1), (2) with physically admissible negativeC – they appear in brane cosmology [16].
When C goes to the range of negative values, the width of the bounce regions steadily increases. As an example, we continue to investigate the massive scalar field case. One of our numerical plots is shown in Fig. 3. We plotted the ϕ = 0 cross-section of bounce regions depending onC. This plot represents a situation, qualitatively different from studied previously for positive or zeroC. Namely, the bounce intervals can merge.
periodic structure described above is restored. On the other hand, initial values from the 1-st interval lead to a very complicated structure of chaotic trajectories. Significant part of them do not fall into a singularity for an arbitrary long time of computer simulations. This fact indicates the presence of a strong chaotic regime, which is qualitatively more complex than the regime corresponding to a rather regular fractal structure described in the previous section. The structure of strong chaos requires further investigations.
5
Less steep potentials
So far we have found that a positive C as well as steep potentials are less favorable for chaos. To prove these dependencies further we consider potentials, less steep than the quadratic one. We will investigate a common family of potentials having power-low asymptotic – Damour– Mukhanov potentials [17]. They were originally introduced to show a possibility for the Universe to have an inflationary expansion without the slow-roll regime. The explicit form of Damour– Mukhanov potential is
V(ϕ) = M
4 0
q "
1 +ϕ
2
ϕ20 q/2
−1 #
. (10)
with three parameters M0,q and ϕ0.
Forϕ≪ϕ0 the potential looks like the massive one with the effective massmeff =M02/ϕ0.
In the opposite case of largeϕit grows like ϕq.
Changing M0 leads only to rescaling a and does not alter the type of chaos. So, we have
a two-parameter (q and ϕ0) family of potentials with different chaotic properties. The main
result of our numerical studies is that the event, described in the previous section – existence of a strong chaos regime – takes place also for this family of gently sloping potentials even with C = 0. Numerical studies show the following picture (see Fig. 4): for a small enough q there exists a corresponding critical value ofϕ0 such that forϕ0 less than the critical one, the strong
chaotic regime exists. Increasingq corresponds to decreasing the critical ϕ0.
Since this regime is absent for quadratic and steeper potentials,qmust at least be less than 2. We can see clearly the strong chaotic regime forq <1.24. Near this value the criticalϕ0decreases
very sharply.
To study further the existence of the strong chaotic regime in gently sloping potentials we consider another family of potentials with the same asymptotic for largeϕand another behavior for small ϕ:
V(ϕ) = M
4 0
q "
1 +ϕ
4
ϕ4 0
q/4
−1 #
Figure 4. The valueϕ0of the potential (10) corresponding to the first merging of the bounce intervals
depending on power indexq.
For small ϕ the potential V(ϕ) ∼ λϕ4 with λ = M04/(4ϕ40), though for large ϕ we still have V(ϕ) ∼ϕq. Our numerical studies have shown that the critical ϕ
0 in this case is close to the
number obtained for Damour–Mukhanov potentials and has the same asymptotic value ∼1.24. We conclude that existence of the strong chaos depends mainly on the parameterq.
So far we found that decreasing steepness of the potential acts as decreasingC. The opposite is also true – if we start from a strong chaos for some suitable gently sloping potential with C = 0, and then allowC to increase, the bounce intervals consecutively separate, strong chaos disappears and with further increasing C the chaotic regime disappears completely [18].
6
Potentials, steeper than quadratic one with negative
C
In the area of potentials steeper than the quadratic one and negativeCwe can see two tendencies acting in opposite directions. We begin with presentation of our numerical results on the strong chaotic regime. We already know that the strong chaos exists forV =m2ϕ2/2 with sufficiently large negative C. On the other hand, our numerical analysis have shown that for the potential V(ϕ)∼ϕ4 there is no strong chaos for any C. In order to find the biggest power index allowing the strong chaos regime, we use the potential family (11). Our numerical data lead to conclusion that large enough negative C could produce the strong chaos forq <2.15.
As for possibility for chaotic regime itself, the value ofC alters the critical ϕ0 in the
expo-nentially steep potentialsV ∼(cosh(ϕ/ϕ0)−1), studied in the Section 3. NegativeCmakes the
chaos possible for a wider range of ϕ0. In the limit C→ −∞the analog of Page point equation
does not depend on C:
√
3V(ϕ)> r
m2P 4π V
′(ϕ)
and, so, the chaos is absent if ϕ0 is less than mP/√12π for any value of C.
7
Conclusions
A
Appendix
In this appendix we briefly describe several dynamical systems which are generalization of (1), (2) in different cosmological scenarios. In all examples we consider onlyC= 0 case.
A.1 Cosmological constant
If we ad a constant termL(often written in the formL= (m2P/8π)Λ, where Λ is the cosmological constant) to a potentialV(ϕ), the cosmological dynamics changes significantly. A new late-time attractor (called the DeSitter regime) with H → p
Λ/3 (and, correspondingly, a ∼ exp(Ht)) appears. Trajectories, reaching this attractor have no maximal expansion point. Large enough Λ leads to a situation when all bouncing trajectories falls into the DeSitter regime, and chaotic behavior disappears [5]. For a massive scalar field this happens if Λ is larger than ∼ 0.3m2.
It should be noticed, that the Page points for the potential V(ϕ) =L+m2ϕ2/2 disappear for
Λ = 0.75m2 [7], so the Page points criterion does not work well in this case.
A.2 Complex scalar f ield
For description of a scalar field with non-zero charge a formalism of complex scalar field is used. The most natural representation of the complex scalar field has the form
φ=xexp(iθ),
where x is the absolute value of the field while θ is its phase. This phase is a cyclical variable corresponding to the conserved quantity – a classical chargeQ≡a3x2θ. The equations of motion˙ are [19]
m2P 16π
¨ a+ a˙
2
2a + 1 2a
+ax˙
2
8 − aV(x)
4 + Q2 4a5x2 = 0
and
¨ x+3 ˙xa˙
a +V
′(x)− 2Q2 a6x3 = 0
with the first integral
− 3 8πm
2
P a˙2+ 1
+a
2
2 ϕ˙
2+ 2V(ϕ) + Q
2
2a4x2 = 0.
A.3 Brane Universe
The equation of motion for a Randall–Sundrum brane Universe [20] in high-energy regime has the form [21]
(5), M(5) is a fundamental 5-dimensional Planck mass, C is an integration
constant.
It is easy to see that in this case the Page points exist for exponential and even steeper than exponential potentials. They disappear only for potentials in the form of infinite potential wall. The critical case now is the potential
V(ϕ) = A (ϕ−ϕ0)2
,
and chaos disappears for A <9M6
(5)/(4π2) [22].
A.4 Anisotropic Universe
The next possibility to generalize equations (1), (2), apart from modifying a scalar field or effective theory of gravity (as have been done in the brane worlds scenario) is to consider a broader than FRW class of metrics. If we lift the assumption of spatial isotropy, the evolution of the Universe will be described by three different scale factors a, b and c. The Einstein equations for a closed Universe with a massive scalar field take the form
( ˙abc)˙
Author is grateful to A.Yu. Kamenshchik, I.M. Khalatnikov, S.V. Savchenko, S.A. Pavluchenko, S.O. Alexeyev, V.O. Ustiansky, P.V. Tretyakov and Parampreet Singh for discussions and col-laboration in studies of chaos in a scalar field cosmology.
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